October 24 and October 31, 2017: Sam Lin "Rank Rigidity in Dimension Three"
Abstract:
Fixing K=-1, 0, 1, a complete Riemannian manifold is said to have higher hyperbolic, Euclidean, or spherical rank if every geodesic admits a normal parallel field making curvature K with the geodesic. Examples for manifolds of higher rank includes locally symmetric spaces. The main goal of the study of rank rigidity is to show that with suitable assumptions, locally symmetric spaces are the only manifolds of higher rank.
Traditionally, rank rigidity has been studied in conjunction with the sectional curvature bounds. After giving a short survey of the historical results, I will talk about how rank rigidity holds in dimension three without a priori sectional curvature bounds.
October 5, 2017: Alexandre Girouard "Discretization and spectrum of compact Riemannian manifolds with boundary"
Abstract: In this talk I will introduce a notion of discretization which allows one to associate a graph to any compact Riemannian manifold with boundary. The eigenvalues of the Dirichlet-to-Neumann map on the manifold are then comparable to those of a natural spectral problem on its discretization. Applications to the construction of a sequence of surfaces with large eigenvalues will be discussed. This is a joint work with Bruno Colbois (Neuchâtel) and Binoy Raveendran.
September 26, 2017: Petra Bonfert-Taylor "Quasiconformal Homogeneity, Active Learning and Programming"
Abstract:
What do the seemingly distinct topics in the title have to do with one another? They are all tied together via the active learning piece. If you don’t know what active learning is, you’ll find out in this talk. Ditto for quasiconformal homogeneity. And while you probably know all about programming, I’ll demonstrate some novel tools we are developing at Thayer to facilitate active learning in a programming class.
Quasiconformal Homogeneity: A quasiconformal homeomorphism between domains is a mapping that behaves as much like a conformal mapping as possible in that, infinitesimally, it distorts spheres at worst into ellipsoids with bounded ratio between major and minor axes. A domain is quasiconformally homogeneous if any two of its points can be mapped onto one another via a quasiconformal homeomorphism of the domain to itself. I’ll speak about geometric and topological constraints that quasiconformal homogeneity imparts on domains (or, more generally, hyperbolic manifolds) and attempts to classify them. The audience will be involved in this presentation.
September 20, 2017: Moon Duchin "Discrete geometry, with applications to voting"
Abstract: Several specialties in mathematics and computer science are built around discrete or metric-space generalizations of classical geometry of manifolds. In math, comparison geometry (with roots in ideas of Alexandrov and others) has taken off in geometric group theory, proving very fruitful for the large-scale study of discrete groups. In CS, there's a burgeoning field of discrete differential geometry, which looks at geometry of meshes and builds up theory of curvature, Laplacians, and so on. I'll survey some ideas in these areas and will explain possible applications to electoral redistricting.
June 7, 2017: Rustam Sadykov "Lusternik-Schnirelmann category of manifolds "
Abstract: The Lusternik-Schnirelmann category of a topological space X is the least number n such that there is an open covering of X by n+1 open subsets contractible to a point in X. I will show that the LS-category of a connected sum of connected orientable manifolds M and N is the maximum of the LS-categories of M and N. This is a joint work in progress with Alex Dranishnikov.
May 31, 2017: Travis Li "Toric log symplectic manifolds"
Abstract: In this talk, we will generalize the toric symplectic manifolds to a new class of Poisson manifolds which are symplectic away from a collection of normal crossing hypersurfaces and also admits a toric symmetry. In analogy with the Delzant classification of toric symplectic manifolds, such manifolds are classified by the decorated log affine polytopes, generalizing the usual Delzant polytope.
May 16, 2017: Patricia Cahn "Linking numbers and Dihedral branched covers of \(S^3 \) and \( S^4 \) "
Abstract: We describe an algorithm for computing the linking numbers between any two rationally null-homologous curves in a 3-fold dihedral cover of $S^3$. This algorithm generalizes an algorithm of Perko, who computed the linking numbers between the two branch curves in the cover. Since every closed oriented 3-manifold is a 3-fold dihedral cover of $S^3$, our algorithm computes the linking number between any two rationally null-homologous curves in any closed oriented 3-manifold. As an application, we explain how this algorithm can be used to compute signatures of dihedral covers of $S^4$ with singular branching sets, using a formula of Kjuchukova. (Joint with Alexandra Kjuchukova).
April 25, 2017: Martin Bridgeman "Simple Length Rigidity for Hitchin Representations"
Abstract: We show that a Hitchin representation is determined by the spectral radii of the images of simple, non-separating closed curves. As a consequence, we classify isometries of the intersection function on Hitchin components of dimension 3 and on the self-dual Hitchin components in all dimensions. This is joint work with Richard Canary, François Labourie
April 11, 2017: Kyler Siegel "Classical Dehn twists are self-diffeomorphisms of surfaces which are supported near closed curves"
Abstract: These generalize to higher dimensions as self-diffeomorphisms of manifolds which are supported near spheres. In fact, Arnold first observed that Dehn twists are natural elements of symplectic geometry, giving rise to self-symplectomorphisms of symplectic manifolds supported near Lagrangian spheres. In recent years these have become very important building blocks in symplectic geometry, especially in light of their connections to algebraic geometry and rich algebraic properties as pioneered by Seidel. In this talk I will describe a strange property of Dehn twists which puts them at the interface between symplectic flexibility and rigidity. As an application, I will construct exotic symplectic manifolds with unexpected properties. Properly understanding these examples requires new deformed versions of holomorphic curve invariants. This is partly based on joint work with Murphy.
April 5, 2017: Renato Bettiol "Non-uniqueness results for the Yamabe and Q-curvature problems"
Abstract: In this talk, I will survey on some recent non-uniqueness results for two central problems in conformal geometry. First, the Yamabe problem of finding metrics with constant scalar curvature in a prescribed conformal class; and, second, its fourth-order analogue called the Q-curvature problem. We show that these problems admit infinitely many pairwise non-homothetic solutions on certain compact and noncompact homogeneous spaces. An essential tool to prove these results is the behavior of eigenvalues of the Laplacian under metric deformations that preserve constant scalar curvature or constant Q-curvature. This is based on separate joint works with Paolo Piccione, Bianca Santoro, and Yannick Sire.
March 28, 2017: Daniel Gardiner "Symplectic embeddings and the Fibonacci numbers"
Abstract: Determining whether or not one symplectic manifold can be embedded into another is a basic problem in symplectic geometry. This question is particularly subtle when the domain and target have the same dimension. For example, McDuff and Schlenk computed when a four-dimensional symplectic ellipsoid can be embedded into a four-dimensional ball, and found that if the ellipsoid is close to round, the answer is partly given by a “infinite staircase” determined by the odd-index Fibonacci numbers, while if the ellipsoid is sufficiently stretched then all obstructions vanish except for the classical volume obstruction. I will report on joint work with Hind and McDuff exploring to what degree analogous results hold in higher dimensions. The methods used by McDuff and Schlenk are explicitly four-dimensional, so this requires developing new techniques and many open questions remain.
February 28, 2017: John Voight "Fake quadrics"
Abstract: A fake quadric is a smooth projective surface that has the same rational cohomology as a smooth quadric surface but is not biholomorphic to one. We complete the classification of fake quadrics uniformized by the product of upper half planes according to the commensurability class of their fundamental group. To accomplish this task, we develop a number of new techniques that explicitly bound the arithmetic invariants of a fake quadric and more generally of an arithmetic manifold of bounded volume arising from a form of $\SL_2$ over a number field. This is joint work with Benjamin Linowitz and Matthew Stover.
February 24, 2017: Sarah Maloni "Combinatorial methods on actions on character varieties"
Abstract: In this talk we consider the SL(2,C)-character variety of the four-holed sphere S, and the natural action of the mapping class group MCG(S) on it. In particular, we describe a domain of discontinuity for the action of MCG(S) on the relative character variety, which is the set of representations for which the traces of the boundary curves are fixed. I will focus on the combinatorial view point we adopt using trace functions on simple closed curves. Time permitting, in the case of real characters, we show that this domain of discontinuity may be non-empty on the components where the relative Euler class is non-maximal. We might also discuss similar results for the three-holed projective plane (and see how these two cases are sort of "dual"). (This is joint work with F. Palesi and S. P. Tan.)
February 21, 2017: Craig Sutton "Detecting the Moments of Inertia of a Molecule via its Rotational Spectrum"
Abstract: Spectral geometry has connections with the field of spectroscopy where one is interested in recovering the structure and composition of a molecule or compound from various spectral data. We demonstrate that the moments of inertia of a molecule can be recovered from its rotational spectrum. Geometrically speaking this means that the isometry classes of left-invariant metrics on $\operatorname{SO}(3)$ can be mutually distinguished via their spectra. In fact, they can be distinguished by their first four heat invariants. More generally, we demonstrate that among compact homogeneous three-manifolds a non-trivial isospectral pair must consist of spherical three-manifolds possessing non-isomorphic cyclic fundamental groups and each is equipped with a so-called Type I metric: at present, no such isospectral pairs exist in the literature. This is joint work with Ben Schmidt (Michigan State University).
February 14, 2017: Maxim Braverman "Equivariant APS index theorem without product structure near the boundary"
Abstract: An APS boundary problem for non–product case was studied by Grubb and Gilkey. In the talk I will discuss an equivariant generalization of this problem and establish a formula for equivariant index of an equivariant Dirac-type operator which is not a product near the boundary. Finally I will use this formula to compute the equivariant eta-invariant of the signature operator on local Kähler-product SKR metric. (Joint work with G. Mashler)
February 7, 2017: Josef Dodziuk "Combinatorial Laplacian and Laplacian on manifolds"
Abstract: The talk will be a survey of some analogies between the Laplacians in these two different settings mainly devoted to applications of facts familiar from PDE and Riemannian geometry to combinatorial Laplacians.
January 31, 2017: Michael Wong "Unoriented skein relations for grid homology and tangle Floer homology"
Abstract: Although the Alexander polynomial does not satisfy an unoriented skein relation, Manolescu (2007) showed that there exists a skein exact triangle for knot Floer homology. In this talk, we will give a combinatorial proof of this result using grid homology. If time permits, we will outline a similar skein relation for the Petkova-Vertesi tangle Floer homology (joint work with Ina Petkova).
January 24, 2017: Carolyn Gordon "Decoding geometry and topology from the Steklov spectrum of orbisurfaces"
Abstract: The Dirichlet-to-Neumann operator of a surface $M$ with boundary or higher-dimensional analogue is a linear map $C^\infty(\partial M)\to C^\infty(\partial M)$ that maps the Dirichlet boundary values of each harmonic function f on M to the Neumann boundary values of f. The spectrum of this operator is discrete and is called the Steklov spectrum. The Dirichlet-to-Neumann operator also generalizes to the setting of orbifolds, e.g., cones. We will address the extent to which the Steklov spectrum encodes the geometry and topology of the surface or orbifold and, in particular, whether it recognizes the presence of orbifold singularities.